Contents Chapter 1. Introduction Chapter 2. Arithmetic-Geometric Preparation 2.1. 2.2. 2.3. 2.4. 2.5. 2.6. 2.7. 2.8. Arithmetic and geometric monodromy groups Distinguished conjugacy classes of inertia generators Branch cycle descriptions The branch cycle argument Weak rigidity Topological interpretation Group theoretic translation of arithmetic exceptionality Remark about exceptional functions over finite fields Chapter 3. Group Theoretic Exceptionality 3.1. 3.2. 3.3. 3.4. 3.5. 3.6. 3.7. 3.8. Notation and definitions Primitive groups General results on exceptionality Examples of exceptionality Nonabelian regular normal subgroups Product action Diagonal action Almost simple groups Chapter 4. Genus 0 Condition 4.1. 4.2. 4.3. 4.4. 4.5. Genus 0 systems in finite permutation groups Diagonal action Product action Almost simple groups Affine action Chapter 5. Dickson Polynomials and Redei Functions Chapter 6. Rational Functions with Euclidean Signature 6.1. 6.2. 6.3. Elliptic Curves Non-existence results Existence results Chapter 7. Sporadic Cases of Arithmetic Exceptionality 7.1. 7.2. 7.3. 7.4. G = C2 x C2 (Theorem 4.13(a)(hi)) G = (Cft) x GL2(3) (Theorem 4.13(c)(1)) G = (Cf) x S3 (Theorem 4.13(c)(ii)) G = (C52) x ((C4 x C2) x C2) (Theorem 4.13(c)(iii)) 1 6 6 6 6 7 8 9 10 10 12 12 13 13 16 19 20 20 22 36 36 39 40 42 46 51 53 53 55 59 70 70 71 72 73 V
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